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PACIFIC JOURNAL OF MATHEMATICS Vol. 216, No. 2, 2004 ALGEBRAIC D-GROUPS AND DIFFERENTIAL GALOIS THEORY Anand Pillay We discuss various relationships between the algebraic Dgroups of Buium, 1992, and diﬀerential Galois theory. In the ﬁrst part we give another exposition of our general differential Galois theory, which is somewhat more explicit than Pillay, 1998, and where generalized logarithmic derivatives on algebraic groups play a central role. In the second part we prove some results with a “constrained Galois cohomological ﬂavor”. For example, if G and H are connected algebraic D-groups over an algebraically closed diﬀerential ﬁeld F , and G and H are isomorphic over some diﬀerential ﬁeld extension of F , then they are isomorphic over some Picard–Vessiot extension of F . Suitable generalizations to isomorphisms of algebraic D-varieties are also given. 1. Introduction We work throughout with (diﬀerential) ﬁelds of characteristic zero. In [8] the notion of a generalized diﬀerential Galois extension (or generalized strongly normal extension) of a diﬀerential ﬁeld was introduced, generalizing Kolchin’s theory of strongly normal extensions, which in turn generalized the Picard–Vessiot theory. The idea was to systematically replace algebraic groups over the constants by “ﬁnite-dimensional diﬀerential algebraic groups”, to obtain new classes of extensions of diﬀerential ﬁelds with a good Galois theory. This idea (almost obvious from the model-theoretic point of view) was implicit in Poizat [11] who gave a model-theoretic treatment of the strongly normal theory. However the “correct” deﬁnition of a generalized diﬀerential Galois extension needed some additional ﬁne-tuning. Nevertheless, our exposition of this general theory in [8] was overly model-theoretic, and possibly remained somewhat obscure to diﬀerential algebraists. We try to remedy this in the current paper by concentrating on the diﬀerential equations that have a good Galois theory, very much in the spirit of Section 7, Chapter IV of Kolchin’s book [4]. The key notion is that of a generalized logarithmic derivative on an algebraic group G over a diﬀerential ﬁeld K (a certain kind of diﬀerential rational map from G to its Lie algebra). We 343 344 ANAND PILLAY will see that such a generalized logarithmic derivative is essentially equivalent to an algebraic D-group structure on G (in the sense of Buium [3]). Our resulting exposition of the generalized diﬀerential Galois theory will be equivalent to that in [8] when the base ﬁeld K is algebraically closed. The general situation (K not necessarily algebraically closed) can be treated using analogues of the V -primitives from [4, IV.10], and we leave the details to others. Let me now say a little more about the generalized logarithmic derivatives, and how they tie up with the Picard–Vessiot/strongly normal theory. Let us ﬁx a diﬀerential ﬁeld K, and assume for now that the ﬁeld CK of constants of K is algebraically closed. A linear diﬀerential equation over K, in vector form, is ∂y = Ay, where y is a n × 1 column vector of unknowns and A is an n × n matrix over K. Looking for a fundamental matrix of solutions, one is led to the equation on GLn : ∂Y = AY , where Y is a n × n matrix of unknowns ranging over GLn , which we can write as ∂(Y )Y −1 = A. Now the map Y → ∂(Y )Y −1 is the classical logarithmic derivative, a ﬁrst-order diﬀerential crossed homomorphism from GLn into its Lie algebra, which is surjective when viewed in a diﬀerentially closed overﬁeld of K. A Picard– Vessiot extension of K for the original equation is then a diﬀerential ﬁeld extension L = K(g), where g ∈ GLn is a solution of ∂(Y )Y −1 = A, and CL = CK . Such an extension exists, and is unique up to K-isomorphism. The group of (diﬀerential) automorphisms of L over K has the structure of an algebraic subgroup of GLn (CK ), and there is a Galois correspondence. In place of GLn one can consider an arbitrary algebraic group G deﬁned over K (not necessarily linear and not necessarily deﬁned over the constants of K). By a generalized logarithmic derivative on G we will mean a ﬁrst-order diﬀerential rational crossed homomorphism µ from G to L(G), deﬁned over K, such that µ is geometrically surjective (that is, surjective when viewed in a diﬀerentially closed overﬁeld) and such that Ker (µ), a ﬁnite-dimensional diﬀerential algebraic subgroup of G, is Zariski-dense in G. The analogue of a linear diﬀerential equation over K will then be an equation (∗) µ(x) = a, where x ranges over G and a ∈ L(G)(K). Under an additional technical condition on the data, analogous to the requirement that the ﬁeld of constants of K be algebraically closed, we can deﬁne the notion of a diﬀerential Galois extension L of K for the equation (∗), prove its existence and uniqueness, identify the Galois group, and obtain a Galois correspondence. In the case where G is deﬁned over the constant ﬁeld CK and µ is the standard logarithmic derivative of Kolchin, we recover Kolchin’s strongly normal extensions (see Theorem 6, Section 7, Chapter IV of [4]). ALGEBRAIC D-GROUPS 345 For G an algebraic group over the diﬀerential ﬁeld K, an algebraic Dgroup structure on G is precisely an extension of the derivation ∂ on K to a derivation on the structure sheaf of G, respecting the group operation. Algebraic D-groups belong entirely to algebraic geometry, and Buium [3] points out that there is an equivalence of categories between the category of algebraic D-groups and the category of ∂0 -groups, ﬁnite-dimensional diﬀerential algebraic groups. The latter category belongs to Kolchin’s diﬀerential algebraic geometry. On the other hand, there is essentially a one-to-one correspondence between algebraic D-group structures on G and generalized logarithmic derivatives on G. So our general diﬀerential Galois theory is in a sense subsumed by the very concept of an algebraic D-group. Details of the above will be given in Sections 2 and 3, including a “Tannakian” approach and an examination of diﬀerent manifestations of the differential Galois group. In Section 4 we will give another relation between algebraic D-groups and the Picard–Vessiot theory: if two algebraic D-groups over an algebraically closed diﬀerential ﬁeld are isomorphic (as D-groups) over some diﬀerential ﬁeld extension of K, then such an isomorphism can be found deﬁned over a Picard–Vessiot extension of K. This uses Kolchin’s diﬀerential Lie algebra, and strengthens the somewhat artiﬁcial results from [9]. We will also give some related results on isomorphisms between algebraic D-varieties, using diﬀerential jets (higher-dimensional versions of diﬀerential tangent spaces). 2. Algebraic D-groups We will brieﬂy describe the algebraic D-groups of Buium in a matter suitable for our purposes. We also introduce the generalized logarithmic derivative on an algebraic group induced by a given D-group structure. We refer to [6] for a discussion of related themes. Let us ﬁx an ordinary diﬀerential ﬁeld (K, ∂). For convenience we also give ourselves a diﬀerential ﬁeld extension (U, ∂) of (K, ∂) that is “universal” with respect to (K, ∂). Namely, U has cardinality κ > |K|, and for any diﬀerential subﬁeld F < U of cardinality < κ and diﬀerential extension L of K of cardinality ≤ κ there is an embedding (as diﬀerential ﬁelds) of L into U over K. CK denotes the ﬁeld of constants of K, and C the ﬁeld of constants of U. We begin by recalling the tangent bundle of an algebraic variety or group over K (in which the derivation on K plays no role). Let X be an algebraic variety over K (maybe reducible). The tangent bundle T (X) of X is another algebraic variety over K, with a canonical surjective morphism π (over K) to X, and is deﬁned locally by equations: ∂P/∂x (x1 , . . . , xn )vi = 0 for P polynomials over K generating the ideal i i of X over K. If X = G is an algebraic group over K, then T (G) has the 346 ANAND PILLAY structure of an algebraic group over K such that the canonical projection to G is an (algebraic) group homomorphism. The group operation on T (G) is obtained by diﬀerentiating the group operation of G. That is, if f (−, −) is the group operation on G and (g, u) and (h, v) are in T (G) then the product (g, u) · (h, v) equals (g · h, dfg,h (u, v)). Note that if λg , ρg denote left and right multiplication by g in G, then we have: (∗) In T (G), (g, u) · (h, v) = (g · h, d(λg )h (v) + d(ρh )g (u)). We will denote T (G)e , the tangent space of G at the identity, by L(G) (for the Lie algebra of G). Then L(G), with its usual vector space group structure, is a normal subgroup of T (G), and we have the exact sequence 0 → L(G) → T (G) → G → {e} of algebraic groups (over K). We denote by i : L(G) → T (G) the natural inclusion, and by π : T (G) → G the canonical surjection above. Note that G acts on L(G) (denoted (g, a) → ag ) by the adjoint map (diﬀerentiating conjugation by g ∈ G at the identity). And this “coincides” with the action of T (G) on the normal subgroup L(G) by conjugation: for a ∈ L(G) and g ∈ G, ag = xax−1 for any x ∈ T (G) such that π(x) = g. For a ∈ L(G), we let la and ra denote the left and right invariant vector ﬁelds on G determined by a. Namely for g ∈ G, la (g) = d(λg )e (a) and ra (g) = d(ρg )e (a). A K-rational splitting of T (G) as a semidirect product of G and L(G) is given by either of the equivalent pieces of data: (a) a K-rational homomorphic section s : G → T (G) (that is π ◦ s = id); (b) a K-rational crossed homomorphism h from T (G) onto L(G) such that h◦i = id. (By deﬁnition α : T (G) → L(G) is a crossed homomorphism if α(xy) = α(x) + xα(y)x−1 .) Note that the set of these K-rational splittings has the structure of a commutative group. For example, using the data in (a), if s1 , s2 are Krational homomorphic sections of the tangent bundle, and si (g) = (g, ui ) for i = 1, 2 then (s1 + s2 )(g) = (g, u1 , +u2 ). The identity element is just the 0-section s0 (g) = (g, 0), which is a K-rational homomorphic section. Let PK (G) denote this commutative group of K-rational splittings of T (G). We denote the crossed homomorphism from T (G) onto L(G) corresponding to the identity of PK (G) by h0 : T (G) → L(G), and the crossed homomorphism corresponding to the homomorphic section s by hs . Remark 2.1. −1 (i) h0 (g, u) = d(ρg )g (u). (ii) More generally, if s is a K-rational homomorphic section of T (G) → G, −1 then hs (g, u) = d(ρg )g (u − s(g)). Proof. This follows directly from formula (∗) for multiplication in T (G). ALGEBRAIC D-GROUPS 347 Let us now bring in the diﬀerential structure. Assume for now that the algebraic variety X is deﬁned over CK the ﬁeld of constants of K. Then it is easy to see that if a ∈ X(U) then working in local coordinates with respect to a given covering of X by aﬃne varieties over CK , (a, ∂(a)) ∈ T (X). If in addition X = G is an algebraic group deﬁned over CK , then the map ∇ : G(U) → T (G)(U) taking g to (g, ∂(g)) is a group embedding. Let lD : G(U) → L(G)(U) be deﬁned by lD(g) = h0 (g, ∂(g)). Then lD is a “diﬀerential rational” crossed homomorphism deﬁned over K, which is precisely Kolchin’s logarithmic derivative. The map lD depends on h0 , and clearly any other h ∈ PK (G) gives rise to another diﬀerential rational crossed homomorphism (over K) from G(U) onto L(G)(U). Remark 2.2. Suppose G is deﬁned over CK . Then: (i) The (standard ) logarithmic derivative lD : G(U) → L(G)(U) is given −1 by lD(g) = d(ρg )g (∂(g)). (ii) lD is surjective. (iii) Ker (lD) is precisely G(C). Proof. (i) follows from Remark 2.1. (ii) Let a ∈ L(G)(U). Then a determines the right invariant vector ﬁeld ra : G(U) → T (G)(U). As U is diﬀerentially closed, the main result of [7] gives g ∈ G(U) such that ∂(g) = ra (g), hence (by (i)), lD(g) = a. (iii) is obvious from (i). Let us now work in a more general context, dropping our assumption that the variety X is deﬁned over CK . Then for a ∈ X(U), (a, ∂(a)) may no longer be a point of T (X) but rather a point of another bundle τ (X) over X, which we now describe. For P (x1 , . . . , xn ) a polynomial over K, let P ∂ denote the polynomial obtained from P by applying ∂ to its coeﬃcients. (So P ∂ = 0 if P is over CK .) Then τ (X) is deﬁned locally by equations: ∂P/∂xi (x1 , . . . , xn )vi + P ∂ (x1 , . . . , xn ) = 0, i where P again ranges over polynomials in the ideal of X over K. (That is, these aﬃne pieces ﬁt together to give an algebraic variety τ (X) over K, together with a canonical projection from τ (X) to X.) It is immediate that (a, ∂a) ∈ τ (X) for a ∈ X(U). It is also immediate that for each a ∈ X, τ (X)a is a principal homogeneous space for the tangent space T (X)a , where the action is addition (with respect to local coordinates above). Moreover this happens uniformly, making τ (X) a torsor for the tangent bundle T (X). In any case, if X is deﬁned over CK , then τ (X) coincides with T (X). By an algebraic D-variety over K we mean a pair (X, s) such that X is an algebraic variety over K and s : X → τ (X) is a regular section deﬁned over K. 348 ANAND PILLAY Now assume again that X = G is an algebraic group over K. Then τ (G) has the structure of an algebraic group over K such that the canonical projection τ (G) → G is a homomorphism. In local coordinates, assuming that multiplication in G is given by the sequence of polynomials f = (fi (x1 , . . . , xn , y1 , . . . , yn ))i=1,...,n over K, then for (g, u), (h, v) ∈ τ (G), the product of (g, u) and (h, v) in τ (G) is given by g · h, df(g,h) (u, v) + (f1∂ (g, h), . . . , fn∂ (g, h)) . We again have the map ∇ : G(U) → τ (G)(U), given (in local coordinates) by ∇(g) = (g, ∂(g)) and this is a group embedding. So we have two K-algebraic groups T (G) and τ (G). Although these need not be isomorphic as algebraic groups, they are “diﬀerential rationally” isomorphic. The following is left to the reader. Lemma 2.3. The map that takes (g, u) to (g, ∂(g) − u) is a (diﬀerential rational) isomorphism of groups between τ (G)(U) and T (G)(U). Although not necessarily rational, it is rational when restricted to the ﬁbers over G. In particular, the above map deﬁnes a K-rational isomorphism between the vector groups τ (G)e and T (G)e = L(G). Note that we have again an exact sequence 0 → τ (G)e → τ (G) → G → e of algebraic groups over K, which by virtue of the (canonical) isomorphism between τ (G)e and L(G) given by Lemma 2.3 can be rewritten as: 0 → L(G) → τ (G) → G → e. Let us again write i for the (canonical) injection of L(G) in τ (G), and π for the canonical surjection τ (G) → G. So if G is deﬁned over CK this agrees with our earlier notation: i : L(G) → T (G) and π : T (G) → G. We can consider splittings (as algebraic groups over K) of τ (G) as a semidirect product of G and L(G). Again each such splitting is determined either by a K-rational homomorphic section s : G → τ (G), or a K-rational crossed homomorphism h : τ (G) → L(G) such that h ◦ i = id on L(G). We will write hs for the crossed homomorphism corresponding to the homomorphic section s, and give explicit formulas below. In any case, we can now deﬁne an algebraic D-group. Deﬁnition 2.4. Let G an algebraic group over K. Then an algebraic Dgroup structure on G over K is precisely a K-rational homomorphic section s : G → τ (G). We write the corresponding algebraic D-group as (G, s). Given an algebraic D-group (G, s) we obtain a generalized logarithmic derivative that we call lDs , a crossed homomorphism (in the obvious sense) from G(U) to L(G)(U): lDs = hs ◦ ∇. Here is the analogue to Remarks 2.1 and 2.2. ALGEBRAIC D-GROUPS 349 Remark 2.5. Let (G, s) be an algebraic D-group over K. (i) (ii) (iii) (iv) −1 For (g, u) ∈ τ (G), hs (g, u) = d(ρg )g (u − s(g)). −1 For g ∈ G(U), lDs (g) = d(ρg )g (∂(g) − s(g)) ∈ L(G)(U). lDs : G(U) → L(G)(U) is surjective. Ker (lDs ) is precisely {g ∈ G(U) : ∂(g) = s(g)}, and is a Zariski-dense subgroup of G(U). Proof. (i) follows from the formula for multiplication in τ (G), and (ii) is an immediate consequence of (i). (iii) Let a ∈ L(G)(U). Again we obtain the right invariant vector ﬁeld ra : G(U) → T (G)(U). Then ra + s : G(U) → τ (G)(U) is also a rational section of τ (G) → G. By [7] there is g ∈ G(U) such that ∂(g) = ra (g)+s(g), hence lDs (g) = a by (i). (iv) Ker (lDs ) is a subgroup of G(U) as lDs is a crossed homomorphism. −1 As d(ρg )g is an isomorphism between T (G)g and T (G)e , we see that Ker (lDs ) is as described in (iii). By [7] for any proper subvariety X of G(U) there is g ∈ G(U) such that ∂(g) = s(g). Hence by (i) Ker (lDs ) is Zariski dense in G. In the context of Remark 2.5, we denote Ker (lDs ) by (G, s) . This is a ﬁnite-dimensional diﬀerential algebraic group, or ∂0 -group in the sense of [3], and is an object belonging to Kolchin’s diﬀerential algebraic geometry. Just for the record, here are some key properties: (G, s) (U) is Zariski-dense in G, the ∂0 -subvarieties of (G, s) are precisely of the form X ∩ (G, s) for X a D-subvariety of (G, s), and for any other algebraic D-group (H, t) the ∂0 -homomorphisms between (G, s) and (H, t) are precisely those induced by algebraic D-group homomorphisms between (G, s) and (H, t). In any case, we have seen that an algebraic D-group structure (G, s) on an algebraic group G over K determines a generalized logarithmic derivative lDs on G. We point out now, just for completeness, that conversely any suitable diﬀerential rational crossed homomorphism map (over K) from G(U) to L(G)(U) determines an algebraic D-group structure on G. Some notation: Let X and Y be algebraic varieties deﬁned over K. By a ﬁrst-order diﬀerential rational map h : X(U) → Y (U), deﬁned over K, we mean a map h from X(U) to Y (U) such that for, for each x ∈ X(U), h(x) ∈ K(x, ∂(x)). Lemma 2.6. Suppose that G is a connected algebraic group over K. Let l : G(U) → L(G)(U) be a ﬁrst-order diﬀerential rational crossed homomorphism, deﬁned over K, which is surjective, and is such that Ker (l) is Zariski-dense in G(U). Then, there are a unique K-rational automorphism σ of L(G), and a unique algebraic D-group structure (G, s) on G (deﬁned over K), such that l = σ ◦ lDs . 350 ANAND PILLAY Proof. By [7], for example, ∇(G(U)) is Zariski-dense in τ (G)(U). Deﬁne l1 on ∇(G(U)) by l1 (x, ∂(x)) = l(x). So by Zariski-denseness, and the properties of l, l1 extends uniquely to a K-rational surjective crossed homomorphism f from τ (G) to L(G). By the Zariski-denseness of Ker (l) in G and the deﬁnition of f , we have π(Ker (f )) = G. On the other hand, dim(τ (G)) = 2 dim(G) and dim(G) = dim(L(G)). Hence dim(Ker (f )) = dim(G). Now π|Ker (f ) : Ker (f ) → G is a group homomorphism so it has ﬁnite kernel. But this ﬁnite kernel is a subgroup of the vector group τ (G)e so has to be trivial. It follows that f |τ (G)e is an isomorphism (over K) with L(G). So there is a unique Kautomorphism σ of L(G) such that (σ ◦ f ) ◦ i is the identity on L(G). Put f1 = σ ◦f . Then f1 gives an algebraic D-group structure (G, s) on G deﬁned over K. For g ∈ G(U), σ ◦ l(g) = f1 (g, ∂(g)) = lDs (g). There is a natural notion of a D-morphism between algebraic D-varieties (X, s) and (Y, t). First note that τ (−) is a functor, so if f : X → Y is a morphism between the algebraic varieties X and Y with everything deﬁned over K then τ (f ) is a morphism, over K, between τ (X) and τ (Y ), again deﬁned over f . (If X, Y, f are deﬁned over the constants, then τ (f ) is just the diﬀerential of f .) In any case, a morphism between algebraic D-varieties X and Y is by deﬁnition a morphism f of algebraic varieties, such that t ◦ f = τ (f ) ◦ s. In particular we extract the notion of an algebraic D-subvariety of (X, s): so an algebraic subvariety Y of X will be the underlying variety of an algebraic D-subvariety of (X, s) if s|Y maps Y to τ (Y ). A homomorphism of algebraic D-groups is a D-morphism that is also a homomorphism of algebraic groups. We call an algebraic D-group (G, s) isotrivial if it is isomorphic over U to a trivial algebraic D-group (H, s0 ), where H is deﬁned over C and s0 is the 0-section of T (G). The interest of the category of algebraic D-groups is that there exist nonisotrivial algebraic D-groups. If A is an abelian variety over U and p : G → A is the universal extension of A by a vector group then G has a D-group structure (deﬁned over the ﬁeld over which A is deﬁned). Moreover any such D-group structure on G is nonisotrivial if A is not isomorphic (as an algebraic group) to an abelian variety deﬁned over C. Given such a Dgroup structure (G, s) on G, p((G, s) ) < A is precisely the Manin kernel of A. This example is worked out in detail in [6]. Let us repeat from [9] the discussion of a nonisotrivial algebraic D-group structure on the commutative algebraic group Gm ×Ga . So let G = Gm ×Ga . Then T (G) = τ (G) can be identiﬁed with {(x, y, u, v) : x = 0} with group structure (x1 , y1 , u1 , v1 )·(x2 , y2 , u2 , v2 ) = (x1 x2 , y1 +y2 , u1 x2 +u2 x1 , v1 +v2 ). ALGEBRAIC D-GROUPS 351 Let s : G → T (G) be the homomorphic section s(x, y) = (x, y, xy, 0). Then (G, s) is an algebraic D-group known to be nonisotrivial. Note that if g = (x, y) ∈ G, then the diﬀerential of multiplication by g −1 at g takes (u, v) ∈ T (G)g to (u/x, v) ∈ L(G). Hence by 2.5, the generalized logarithmic derivative lDs corresponding to s is: lDs (x, y) = ((∂(x) − xy)/x, ∂(y)) = (∂(x)/x − y, ∂(y)). In particular (G, s) = Ker (lDs ) = {(x, y) ∈ G : ∂(x)/x = y, ∂(y) = 0}, which can be identiﬁed with the subgroup of Gm deﬁned by the second-order equation ∂(∂(x)/x) = 0. 3. Diﬀerential Galois theory The conventions of the previous section are in force. In particular (K, ∂) is a diﬀerential ﬁeld of characteristic 0, and we may refer also to the universal diﬀerential ﬁeld extension (U, ∂) of K. By a logarithmic diﬀerential equation over a diﬀerential ﬁeld (K, ∂) we mean something of the form (∗) lDs (x) = a, where (G, s) is an algebraic D-group deﬁned over K and a ∈ L(G)(K). (So the indeterminate x ranges over G.) As remarked in the introduction, a special case is the equation ∂(X) = AX, where X ranges over GLn and A is an n × n matrix over K. In order to give the right notion of a diﬀerential Galois extension of K for (∗), we need to place a further restriction on K, G and s. Deﬁnition 3.1. Suppose (G, s) is an algebraic D-group over K. We say that (G, s) is K-large, if for every (maybe reducible) algebraic D-subvariety X of G, which is deﬁned over K, X(K) ∩ (G, s) is Zariski-dense in X. Remark 3.2. (i) The intuitive meaning of (G, s) being K-large is that (G, s) has enough points with coordinates in K. (ii) Suppose that G is deﬁned over CK and that s = s0 , the 0-section of T (G). Then, if CK is algebraically closed, (G, s) is K-large. In the next remark, we refer to diﬀerential closures of K. A diﬀerential closure of K is a diﬀerential ﬁeld extension of K that embeds over K into any diﬀerentially closed ﬁeld containing K. The diﬀerential closure of K is unique up to K-isomorphism, and is written as K̂. Kolchin calls K̂ the constrained closure of K. Remark 3.3. (G, s) is K-large if and only (G, s) (K) = (G, s) (K̂) for some (any) diﬀerential closure K̂ of K. 352 ANAND PILLAY Proof. Assume ﬁrst that (G, s) (K) = (G, s) (K̂). Let X be a D-subvariety of (G, s) deﬁned over K. The irreducible components X1 , . . . , Xr of X are deﬁned over K so also K̂ and are also D-subvarieties of (G, s). By [7] for example for any nonempty Zariski open subset of any Xi deﬁned over K̂, there is a ∈ U (K̂ such that ∂(a) = s(a). By our assumptions a ∈ (G, s) (K). So X ∩ (G, s) (K) is Zariski-dense in X. Conversely, suppose (G, s) is K-large. Let a ∈ (G, s) (K̂), and suppose for a contradiction that a ∈ / G(K). Now a is constrained over K in the sense of Section 10, Chapter III of [4]. This means that there is some diﬀerential polynomial P (x) over K such that P (a) = 0, and whenever b is a diﬀerential specialization of a over K and P (b) = 0 then b is a “generic” specialization of a over K (in model-theoretic language tp(b/K) = tp(a/K)). Let X be the irreducible K-subvariety of G whose generic point is a. Then (as ∂(a) = s(a)), X is a D-subvariety of (G, s). Moreover the diﬀerential specializations of a over K are precisely those b such that b ∈ X and ∂(b) = s(b). Now subject to the conditions “x ∈ X and ∂(x) = s(x)”, the condition P (x) = 0 is clearly equivalent to x ∈ Y for Y some proper subvariety of X deﬁned over K. By our assumptions there is b ∈ G(K) such that b ∈ X \ Y and ∂(b) = s(b). This is a contradiction. We call a diﬀerential ring (R, ∂) simple if it has no proper nontrivial diﬀerential ideals. We refer to [12] for a discussion of simple diﬀerential rings. Lemma 3.4. Suppose that R is a simple diﬀerential ring over K that is ﬁnitely generated over K. Then R embeds over K into some diﬀerential closure of K. Proof. As R has no zero-divisors, R embeds in U over K. Let R = K[a]∂ be diﬀerentially generated over K by the ﬁnite tuple a. Suppose that π(a) = b is a diﬀerential specialization over K. Then π extends to a surjective ring homomorphism π : R → K[b]∂ . The kernel is a diﬀerential ideal, so must be trivial. Thus b is a generic specialization of a over K. It follows that a is constrained over K so lives in some diﬀerential closure of K, as does R. We can now give the main deﬁnition. Deﬁnition 3.5. Let (G, s) be a K-large algebraic D-group deﬁned over K, and lDs (x) = a be a logarithmic diﬀerential equation over K for (G, s). By a diﬀerential Galois extension of K for the equation lDs = a we mean a diﬀerential ﬁeld extension L of K of the form K(α) for some solution α of the equation such that K[α], the (diﬀerential) ring generated by K and the coordinates of α, is a simple diﬀerential ring. Lemma 3.6 (Existence and uniqueness of diﬀerential Galois extensions). If (G, s) is a K-large algebraic D-group deﬁned over K, and a ∈ L(G)(K), ALGEBRAIC D-GROUPS 353 then there exists a diﬀerential Galois extension L of K for the equation lDs (x) = a. Moreover, any two such extensions are isomorphic over K as diﬀerential ﬁelds. Proof. By Remark 2.5 (iii) there is a solution β ∈ G(U) of lDs (x) = a. Let α be a maximal diﬀerential specialization of β over K. Then K[α] is a simple diﬀerential ring and α is also a solution of lDs = a, so we get existence. Let L = K(α). Suppose L1 is another diﬀerential Galois extension of K for the equation, generated by the solution γ say. By Lemma 3.4 we may assume that both L and L1 are contained in some diﬀerential closure K̂ of K. By Remark 3.3, (G, s) (K̂) = (G, s) (K). As both α and γ are solutions of lDs (x) = a, it follows that α−1 · γ ∈ (G, s) (K̂) = (G, s) (K). Thus clearly L = L1 . Here are some alternative characterizations of diﬀerential Galois extensions: Lemma 3.7. Let (G, s) be a K-large algebraic D-group over K, and L = K(α) a diﬀerential ﬁeld extension of K, where α is a solution of lDs (x) = a (with a ∈ L(G)(K)). Then the following are equivalent: (i) L and α satisfy Deﬁnition 3.5. (ii) L is contained in some diﬀerential closure of K. (iii) (G, s) (K) = (G, s) (L) and (G, s) is L-large. Proof. (i) implies (ii) is given by Lemmas 3.4 and 3.6 and (ii) implies (iii) follows from Remark 3.3. (iii) implies (i). Assume L satisﬁes (iii). Then using Remark 3.3, we have that (G, s) (L̂) = (G, s) (K). Now K̂ embeds in L̂ over K, hence by Lemmas 3.4 and 3.6, there is a solution β ∈ G(L̂) of lDs (x) = a, such that K[β] is a simple diﬀerential ring. Now α = β · g for some g ∈ (G, s) (K), so clearly K[α] is also a simple diﬀerential ring. Condition (iii) is the analogue of “no new constants” in the strongly normal case. Remark 3.8. Suppose K is algebraically closed. Then the diﬀerential Galois extensions of K in the sense of Deﬁnition 3.5 coincide with the generalized strongly normal extensions of K in the sense of [8]. In particular, L is a strongly normal extension of K (in the sense of Kolchin) just if L is a diﬀerential Galois extension of K for an equation lDs (x) = a, on an algebraic D-group (G, s), where G is deﬁned over CK and s = s0 . Proof. This follows from Proposition 3.4 of [8]. We now point out that given L as above, Aut∂ (L/K) has the structure of a diﬀerential algebraic group (over K) in two diﬀerent ways. One corresponds to the usual diﬀerential Galois group in the linear case, and is simply of the 354 ANAND PILLAY form (H, s) (K), where H is an algebraic D-subgroup of (G, s). The other, corresponding to the “intrinsic” Galois group introduced by Katz, is of the form (H1 , s1 ) (L) for s1 another algebraic D-group structure on G (deﬁned over K), and H1 a D-subgroup of (G, s1 ) deﬁned over K. The algebraic D-groups (H, s) and (H1 , s1 ) will be isomorphic, but not necessarily over K, unless they are commutative. (But of course if K is algebraically closed, H and H1 will be isomorphic over K as algebraic groups.) So ﬁx L = K(α) as in Deﬁnition 3.5. By 3.7 we have L < K̂ for some ﬁxed copy of the diﬀerential closure of K. Aut∂ (L/K) denotes the group of diﬀerential ﬁeld automorphisms of L over K. Note that for any σ ∈ Aut∂ (L/K), σ(α) is also a solution of lDs = a, hence σ(α) = b · c(σ) for a unique c(σ) ∈ (G, s) (L) = (G, s) (K) (= (G, s) (K̂)). Lemma 3.9. The map c above is a group isomorphism between Aut∂ (L/K) and (H, s) (K) for some algebraic D-subgroup H of G deﬁned over K. Proof. As an automorphism σ of L over K is determined by its action on α, clearly the map is a group isomorphism with its image. So all that we need is that the image is of the required form. The model-theoretic proof of this goes through showing that the image is a deﬁnable subgroup of (G, s) (K̂), and then using quantiﬁer-elimination for diﬀerentially closed ﬁelds. We will give an algebraic proof, after ﬁrst discussing the second incarnation of the Galois group. First, the equation lDs (x) = a equips G with another structure of an algebraic D-variety (but not in general an algebraic D-group). Let ra be the right invariant vector ﬁeld on G determined by a on G. So s + ra is a K-rational section of τ (G) → G, which we denote by s . Note that the equation lDs (x) = a on G is equivalent to the equation ∂(x) = s (x). Now G acts on itself by left translation. Let S < G be the intersection of the stabilizers of the algebraic D-subvarieties of the algebraic D-variety (G, s ) that are deﬁned over K: namely (working in some algebraically closed ﬁeld containing K), S = {g ∈ G : g · X = X for all D-subvarieties X of (G, s ) deﬁned over K}. S is an algebraic subgroup of G deﬁned over K. S is precisely the analogue in our context of the intrinsic diﬀerential Galois group introduced by Katz in the Picard–Vessiot case and discussed by Bertrand in [1]. In fact G can be naturally equipped with another structure of an algebraic D-group, (G, s1 ). Deﬁne the section s1 : G → τ (G) by s1 (g) = s(g) + ra (g) − la (g). Lemma 3.10. (G, s1 ) is an algebraic D-group, and the action of G on itself by left multiplication is an action of the algebraic D-group (G, s1 ) on the algebraic D-variety (G, s ). Moreover S is an algebraic D-subgroup of (G, s1 ). ALGEBRAIC D-GROUPS 355 Proof. An easy computation. Note that it follows that (G, s1 ) (U) acts on (G, s ) (U). Let Z be the set of solutions of lDs (x) = a in G(L). Note that Z is precisely (G, s ) (L). In any case, Z is a principal homogeneous space for (G, s) (K) (acting on the right). As (G, s) (K) = (G, s) (K̂), and L < K̂, X is also the solution set of lDs (x) = a in G(K̂). Clearly Aut ∂ (L/K) acts on Z and any σ ∈ Aut ∂ (L/K) is determined by its action on Z. In fact, as L = K(β) for any β ∈ Z, σ ∈ Aut ∂ (L/K) is determined by any pair (β, σ(β)) for β ∈ Z. Lemma 3.11. The action of Aut ∂ (L/K) on Z is isomorphic to the action (by left multiplication) of (S, s1 ) (L) on Z: The isomorphism d say takes σ to σ(β) · β −1 for some (any) β ∈ Z. Proof. We know that (S, s1 ) acts on Z by left multiplication (by Lemma 3.10). Suppose σ ∈ Aut ∂ (L/K). Let d(σ) ∈ G(L) be such that σ(α) = d(σ) · α. As σ(α) ∈ Z, we have by Lemma 3.10 that d(σ) ∈ (G, s1 ) (L). As any β ∈ Z is of the form α · c for c ∈ (G, s) (L) = (G, s) (K) and σ ﬁxes K pointwise, we see that (∗) σ(β) = d(σ) · β for all β ∈ Z. We only have to see that d(σ) ∈ S. Let X be any D-subvariety of (G, s ) deﬁned over K. As K̂ is diﬀerentially closed and Z = (G, s ) (K̂), it follows that Z ∩ X is Zariski-dense in X. For β ∈ Z ∩ X, σ(β) ∈ Z ∩ X too. By (∗) and Zariski-denseness, d(σ) · X = X. Thus d(σ) ∈ S. Conversely, suppose that g ∈ (S, s1 ) (L). Then g · α ∈ Z. Let X be the algebraic variety deﬁned over K whose K-generic point is α. Then, X is a D-subvariety of (G, s ). Thus g · X = X and so g · α ∈ X, and ∂(g · α) = s (g · α). Hence g · α is a diﬀerential specialization of α over K. By simplicity of K[α], α and g · α satisfy exactly the same diﬀerential polynomial equations over K. As both α and g · α generate L it follows that there is an automorphism σ of L over K such that σ(α) = g · α. So g = d(σ). Conclusion of proof of Lemma 3.9. Let H be the image of S under conjugation by α−1 in G (g → α−1 · g · α). Then H is an algebraic D-subgroup of (G, s). By 3.11 and what we already know about 3.9, the image of the embedding c : Aut ∂ (L/K) → (G, s) (K) is precisely (H, s) (K). As the latter is Zariski-dense in H, H is also deﬁned over K. Let us ﬁx the isomorphism c between Aut ∂ (L/K) and (H, s) (K) = (H, s) (K̂). Lemma 3.12. There is a Galois correspondence between the set of diﬀerential ﬁelds in between K and L and the set of algebraic D-subgroups of 356 ANAND PILLAY (H, s) deﬁned over K (equivalently deﬁned over K̂): given K < F < L the corresponding group is HF the Zariski closure of the set of h ∈ (H, s) (K) such that c−1 (h) is the identity on F . Proof. This is Theorem 2.12 of [8] after making the translation between deﬁnable subgroups and D-subgroups. Let us complete this section with an example. We will consider the nonisotrivial algebraic D-group structure (G, s) on Gm × Ga discussed at the end of Section 2, and exhibit a (natural) diﬀerential Galois extension K < L whose diﬀerential Galois group is (G, s) (or rather (G, s) (K)). In fact there will be an intermediary diﬀerential ﬁeld K < F < L such that each of K < F and F < K are Picard–Vessiot extensions, but K < L is not a Picard–Vessiot extension. Recall that s : G → T (G) is given by s(x, y) = (x, y, xy, 0), and lDs : G → L(G) is lDs (x, y) = (∂(x)/x−y, ∂(y)), and so a logarithmic diﬀerential equation on (G, s) has the form {∂(x)/x − y = a1 , ∂(y) = a2 }. If we restrict our attention to equations where a1 = 0, we obtain equations of the form ∂(∂(x)/x) = a on Gm . Also (G, s) can be identiﬁed with {x ∈ Gm : ∂(∂(x)/x)) = 0}. Note that the embedding x → (x, 0) of Gm in G and surjection (x, y) → y of G onto Ga induces an exact sequence 0 → (Gm , (s0 )Gm ) → (G, s) → (Ga , (s0 )Ga ) → 0 of algebraic D-groups, where s0 denotes the 0-sections for the corresponding groups. The important fact is that (G, s) is not a product (as a D-group) of the two groups, which is the reason that (G, s) is nonisotrivial. We will take the ground ﬁeld of constants to be C. Let K = C(ect : c ∈ C) 2 and L = K(t, et ). So L is a subﬁeld of a ﬁxed diﬀerential closure Ĉ = K̂ = L̂ of C. Lemma 3.13. (G, s) is K-large. Proof. It is enough to show that all solutions of ∂(∂(x)/x) = 0 in K̂ are in K, that is all solutions of ∂(d)/d = c for c ∈ C that are in K̂ are in K. But this is clear, because ect is one such such solution and the others are obtained by multiplying by a constant. Lemma 3.14. L is a diﬀerential Galois extension of K for the equation ∂(∂(x)) = 2 . The Galois group is (G, s) (K). 2 Proof. Note that L is generated over K as a diﬀerential ﬁeld by et , which is a solution of ∂(∂(x)) = 2. To show that the Galois group is as stated, it is enough to show that tr.deg(L/K) = 2, which is well-known. Note that K(t) is a Picard–Vessiot extension of K, and L is a Picard– Vessiot extension of K(t), but L is not a Picard–Vessiot extension of K. ALGEBRAIC D-GROUPS 357 4. Isomorphisms of algebraic D-groups and algebraic D-varieties We will prove: Proposition 4.1. Suppose (K, ∂) is an algebraically closed diﬀerential ﬁeld, (G, s) and (H, t) are connected algebraic D-groups over K and there is an isomorphism f between (G, s) and (H, t) deﬁned over some diﬀerential ﬁeld extension of K. Then there is such an isomorphism deﬁned over a Picard– Vessiot extension of K. Here is a restatement of the theorem in the language of Kolchin’s constrained cohomology (see [5]). Corollary 4.2. Suppose (K, ∂) has no proper Picard–Vessiot extensions. Then for any connected ∂0 -group G deﬁned over K, Hc1 (Aut ∂ (K̂/K), G(K̂)) is trivial. In Proposition 3.11 of [9], Corollary 4.2 was stated in the special case that H(K) = H(K̂) (which amounts to saying that the corresponding algebraic D-group is K-large). Kolchin’s diﬀerential tangent space and its properties (see Chapter 8 of [5]) play an important role in the proof of Proposition 4.1. We will summarize the key properties (in the language of algebraic D-groups). Recall ﬁrst that if V is a ﬁnite-dimensional vector space over U, then a ∂-module structure on V is an additive map DV : V → V such that DV (av) = ∂(a)v + aDV (v) for all a ∈ U and v ∈ V . V ∂ denotes {v ∈ V : DV (v) = 0}, a vector space over C with C-dimension the same as the U-dimension of V Fact 4.3. Suppose (G, s) is a connected algebraic D-group deﬁned over K, and V = L(G) is its Lie algebra (tangent space at the identity). Then: (i) s equips V with a canonical ∂-module structure DV , deﬁned over K. (ii) For any automorphism f of (G, s), dfe ∈ GL(V ) restricts to a C-linear automorphism of V ∂ , and moreover f is determined by dfe |V ∂ . Proof of Proposition 4.1. First (G, s) and (H, t) will be isomorphic over K̂. Let f be such an isomorphism. Let c be a ﬁnite tuple from K̂ generating the smallest ﬁeld of deﬁnition of f . Write f as fc . Let (V, DV ) be as in Fact 4.3 for (G, s). Let d be a C-basis for V ∂ contained in K̂. Let L = Kc, d be the diﬀerential ﬁeld generated by c and d over K. Claim I. L is a strongly normal extension of K. Proof. As L < K̂, CL = CK . We have to show that for any automorphism σ of the diﬀerential ﬁeld U ﬁxing K pointwise, σ(L) is contained in the diﬀerential ﬁeld generated by L and C. First σ(d) is another basis of V ∂ , so with respect to the basis d we may write σ(d) as a n × n nonsingular matrix B ∈ GLn (C). On the other hand fσ (c) = fc · h for a unique automorphism 358 ANAND PILLAY h of (G, s). By Fact 4.3(ii), h is determined by dhe |V ∂ , which by in terms of the basis d, is another nonsingular n × n matrix A over C. It follows that (σ(c), σ(d)) is in the diﬀerential ﬁeld generated by K, c, d, A and B. In particular σ(L) ⊆ LC . Claim II. L is a Picard–Vessiot extension of K. Proof. Let σ ∈ Aut ∂ (L/K), and write the matrices A, B (which will be in GLn (CK )) as Aσ , Bσ . Then it is easy to check that σ → (Aσ , Bσ ) is an embedding of Aut ∂ (L/K) into GLn (CK ) × GLn (CK ). From Kolchin’s characterizations of Picard–Vessiot extensions, we get Claim II. As f is deﬁned over L, we have proved Proposition 4.1. Finally let us give some additional results with a similar ﬂavor. The ﬁrst is really just a remark and is related to the themes of Buium’s book [2]. Buium calls an algebraic D-variety (X, s) split if (X, s) is isomorphic (over U) to a trivial algebraic D-variety, namely one of the form (Y, s0 ), where Y is deﬁned over C and s0 is the 0-section. He proves that any D-variety (X, s) such that X is a projective variety is split. Moreover assuming Y deﬁned over algebraically closed K, then (Y, s) is split over some strongly normal extension. Remark 4.4. Let K be a diﬀerential ﬁeld with algebraically closed constant ﬁeld. Suppose that (X, s) is an algebraic D-variety over K that is split. Then (X, s) is split over some strongly normal extension K1 of K. Proof. We can ﬁnd an isomorphism f of (X, s) with some trivial (Y, s0 ) deﬁned over K̂. Let again f = fc with c the smallest ﬁeld of deﬁnition of f . Let K1 = Kc . Then, as K1 is contained in K̂ and CK̂ = CK , we have that CK1 = CK . For any (diﬀerential) automorphism σ of U, Then fσ(c) = fc ◦ g for some automorphism g of (Y, s0 ). But then g must be deﬁned over C. Hence σ(c) is rational over K(c)(C), so σ(K1 ) ⊆ K1 C . This shows that K1 is a strongly normal extension of K. The next result is a generalization of Proposition 4.1 in which the higherdimensional versions of diﬀerential tangent spaces from [10] enter the picture. We use freely the results from that paper. Proposition 4.5. Suppose that K has algebraically closed constant ﬁeld. Let (X, s) and (Y, t) be algebraic D-varieties deﬁned over K. Suppose that a ∈ (X, s) (K), b ∈ (Y, t) (K) are nonsingular points on X, Y respectively, and that there is some isomorphism f between (X, s) and (Y, t) such that f (a) = b. Then there is such an isomorphism deﬁned over a Picard–Vessiot extension of K. ALGEBRAIC D-GROUPS 359 Proof. For each m ≥ 1, let Vm the the U-vector space M/Mm+1 , where M is the maximal ideal of the local ring of X at a. Vm is deﬁned over K. For any automorphism h of X such that h(a) = a, h induces a linear automorphism hm say of Vm . Moreover if h is another automorphism of X ﬁxing a, then h = h if and only if hm = hm for all m. So far nothing has been said about the D-variety structure. As a ∈ (X, s) , ∂ extends to a derivation ∂ on the local ring of X at a that preserves M and all its powers. This gives Vm the structure of a ∂ module (Vm , DVm ) over U, deﬁned over K (for all m). If h is an automorphism of the algebraic D-variety (X, s) that ﬁxes a then hm ∈ GL(Vm ) restricts to a C-linear automorphism h∂m of GL(Vm∂ ) (where Vm∂ is the solution space of DVm = 0). Moreover hm is determined by h∂m . Now we may ﬁnd an isomorphism f between (X, s) and (Y, t) such that f (a) = b and f is deﬁned over K̂. Let c generate the smallest ﬁeld of deﬁnition of f . So c is a ﬁnite tuple from K̂. Write f as fc . For any diﬀerential automorphism σ of U that ﬁxes K pointwise, σ(fc ) = fσ(c) is also an isomorphism of (X, s) with (Y, t) taking a to b. Hence fσ(c) ◦ fc−1 is an automorphism of (X, s) taking a to itself. Write hσ for this map. Claim I. There is m such that for all σ, τ ∈ Aut ∂ (U/K), hσ = hτ if and only if (hσm )∂ = (hτm )∂ . Proof. This follows from compactness and the earlier remarks as the set of hσ is a uniformly deﬁnable family of automorphisms of (X, s). Now let m be as in Claim I and let d be a C-basis for (Vm )∂ . Claim II. K1 = Kc, d is a strongly normal extension of K. Proof. Let σ ∈ Aut ∂ (U/K). As σ(d) is also a basis for (Vm )∂ , σ(d) ∈ Kd C . On the other hand, by virtue of d, (Vm )∂ can be identiﬁed with C r for suitable r, and thus (hσm )∂ can be identiﬁed with an element of GLr (C). By Claim I, it follows that σ(c) ∈ Kc, d C . Thus σ(K1 ) ⊆ K1 C . As in the proof of 4.1, we conclude that actually K1 is a Picard–Vessiot extension of K. As c ∈ K1 , this gives the proposition. Acknowledgements. This work was carried out while the author was an Invited Professor at the University of Paris VI in June 2003, and while visiting the University of Naples II in July 2003. I am indebted to Daniel Bertrand at Paris VI for his hospitality, many helpful conversations, and his interest in the work reported here. Thanks also to Zoe Chatzidakis of University Paris VII-CNRS for her suggestions, and to Paola D’Aquino of Naples II for her hospitality. 360 ANAND PILLAY References [1] D. Bertrand, Review of “Lectures on diﬀerential Galois theory”, by A. Magid, Bull. Amer. Math. Soc., 33 (1996), 289–294. [2] A. 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Pillay, Diﬀerential Galois theory I, Illinois J. Math., 42 (1998), 678–699, MR 1649893 (99m:12009), Zbl 0916.03028. [9] A. Pillay, Finite-dimensional diﬀerential algebraic groups and the Picard–Vessiot theory, in ‘Diﬀerential Galois Theory’, Banach Center Publications, 58, Polish Academy of Sciences, 2002, 189–199, MR 1972454 (2004c:12008), Zbl 1036.12006. [10] A. Pillay and M. Ziegler, Jet spaces of varieties over diﬀerential and diﬀerence ﬁelds, Selecta Math., 9(4) (2003), 579–599, MR 2031753. [11] B. Poizat, A Course in Model Theory. An Introduction to Contemporary Mathematical Logic, Universitext, Springer, New York, 2000, MR 1757487 (2001a:03072), Zbl 0951.03002. [12] M. van der Put and M. Singer, Galois Theory of Linear Diﬀerential Equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer, Berlin, 2003, MR 1960772 (2004c:12010), Zbl 1036.12008. Received January 20, 2004 and revised May 7, 2004. The author was supported by NSF grant DMS 0300639 and NSF Focused Research Grant DMS 0100979. Department of Mathematics University of Illinois at Urbana-Champaign Urbana, Illinois 61801-2975 E-mail address: [email protected] This paper is available via http://www.pacjmath.org/2004/216-2-8.html.